Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown below.
 


To determine the amplitude, midline, period, and an equation involving the sine function for the given graph, follow these steps:

1. **Amplitude (A)**: The amplitude is the maximum distance from the midline to the peak of the graph. Given the information:
   \[
   A = 2
   \]

2. **Midline (y = D)**: The midline is the horizontal line that runs through the middle of the graph. This is the average of the maximum and minimum values of the sine function. If the midline value is \(y = 1\), then:
   \[
   y = 1
   \]

3. **Period (P)**: The period is the length of one complete cycle of the sine wave on the x-axis. The period is often determined by the distance between two successive peaks or two successive troughs. Suppose the period of the graph is \(4\), then:
   \[
   P = 4
   \]

4. **Equation of the Sine Function**: The general form of the sine function is:
   \[
   y = A \sin(Bx + C) + D
   \]
   Where:
   - \(A\) is the amplitude.
   - \(D\) is the midline value.
   - \(B\) is related to the period by the formula \(B = \frac{2\pi}{P}\).
   - \(C\) is the phase shift (we will assume no phase shift unless specified).

Given:
- Amplitude \(A = 2\)
- Midline \(y = 1\)
- Period \(P = 4\)

Calculate \(B\):
\[
B = \frac{2\pi}{P} = \frac{2\pi}{4} = \frac{\pi}{2}
\]

Assuming there is no phase shift (\(C = 0\)), the equation of the sine function is:
\[
y = 2 \sin\left(\frac{\pi}{2}x\right) + 1
\]

So, the exact equation is:
\[
y = 2 \sin\left(\frac{\pi}{2}x\right) + 1
\]

### Summary
- **Amplitude**: \(2\)
- **Midline**: \(y = 1\)
- **Period**: \(4\)
- **Equation**: \(y = 2 \sin\left(\frac{\pi}{2}x\right) + 1\)

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Do you have any specific questions or need further details?

Here are 8 related questions to expand on the information:
1. What is the exact value of the phase shift (C) if there is any?
2. How would the graph look if the amplitude was 3 instead of 2?
3. What happens to the period if the frequency of the sine function is doubled?
4. Can you determine the maximum and minimum points of the function?
5. How would the graph be affected if the midline was at \(y = 0\)?
6. What is the effect of adding a phase shift to the function?
7. How can you verify the period of the sine function from the graph?
8. What would the equation be if the function was a cosine function instead of sine?

**Tip:** To determine the midline of a sine function graphically, find the average of the maximum and minimum values of the function. This helps to identify the central axis around which the sine function oscillates.

Learn how to determine the amplitude, midline, period, and equation involving the sine function from a given graph. This problem involves trigonometric concepts such as amplitude, midline, and period, and how to construct the sine function equation based on these parameters. Suitable for high school students studying trigonometry.

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